Nonlinear PDEs
Mathematical challenges
Nonlinear partial differential equations provide the continuum language for systems whose response depends on their current state. We study nonlinear diffusion, conservation laws and gradient-flow structures, especially when diffusion is degenerate or coefficients are irregular.
Central questions concern existence, uniqueness and regularity of solutions, the propagation of singularities, and the way qualitative structures survive approximation and limiting procedures. Variational methods and Γ-convergence help us pass between microscopic models, effective energies and macroscopic equations. These ideas also prepare the analytical foundations for stochastic models: identifying robust estimates and compactness principles is essential when deterministic equations are perturbed by fluctuations.
Current directions
- Gradient flows and variational structures
- Conservation laws and degenerate diffusion
- PDEs with irregular coefficients
- Regularity and singular limits
- Γ-convergence and effective equations
Related people
Related projects
Selected publications
-
SVI solutions to stochastic nonlinear diffusion equations on general measure spaces
Published · Journal of evolution equations, 24 (2024) 4, p. 94. We establish a framework for the existence and uniqueness of solutions to stochastic nonlinear (possibly multi-valued) diffusion equations driven by multiplicative noise, with the drift operator $L$ being the generator of a…
-
Optimal Regularity in Time and Space for Nonlocal Porous Medium Type Equations
A broad class of possibly non-unique generalized kinetic solutions to hyperbolic-parabolic PDEs is introduced. Optimal regularity estimates in time and space for such solutions to nonlocal, and spatially inhomogeneous variants of the porous medium equation are shown in the scale of Sobolev…
-
Solutions to the stochastic thin-film equation for initial values with non-full support
Published · Transactions of the American Mathematical Society, 379 (2026) 4, pp. 2343-2383. The stochastic thin-film equation with mobility exponent $n\in [\frac{8}{3},3)$ on the one-dimensional torus with multiplicative Stratonovich noise is considered. We show that martingale solutions exist for non-negative initial values. This advances on…
-
The Porous Medium Equation: Large Deviations and Gradient Flow with Degenerate and Unbounded Diffusion
Published · Communications on pure and applied mathematics, 78 (2025) 9, pp. 1609-1655. The problem of deriving a gradient flow structure for the porous medium equation which is {\em thermodynamic}, in that it arises from the large deviations of some microscopic particle system, is studied…
-
Long-time behaviour of stochastic Hamilton-Jacobi equations
Published · Journal of functional analysis, 286 (2024) 4, p. 110269. The long-time behavior of stochastic Hamilton-Jacobi equations is analyzed, including the stochastic mean curvature flow as a special case. In a variety of settings, new and sharpened results are obtained. Among them…
-
Lyapunov exponents and synchronisation by noise for systems of SPDEs
Published · The annals of probability, 52 (2024) 5, pp. 1903-1953. Quantitative estimates for the top Lyapunov exponents for systems of stochastic reaction-diffusion equations are proven. The treatment includes reaction potentials with degenerate minima. The proof relies on an asymptotic expansion of the…